Projection Biases Perceived Angles on a Picture Surface
Igor Juricevic, Shazma Rajani, Sherief Hammad, & John M. Kennedy
Abstract
Angles on picture surfaces may be perceived incorrectly.
This illusion may be caused by projective information.
Alternatively, the angles may only be misperceived if they
are depicting very standard angles such as 90°. Two
experiments used parallel perspective to test these
hypotheses. In Experiment 1, subjects judged angles
presented alone (i.e., as V-shapes) or as parts of a rightangled, scalene triangle depicted as the top surface of a
double-cube (depicting angles of 26.6°, 63.4°, and 90°). The
largest errors were in the double-cube condition. They were
biased towards each of the depicted angles of the doublecube. In Experiment 2 the V-shapes were shown as parts of
triangles, to test whether triangles alone could create the
illusion, perhaps by providing projective 3-D information for
a single triangle being depicted at different tilts. The errors
were indeed biased towards the depicted angles, but were
less than in the double-cube condition. The results support
a projective theory of the illusion: that the foreshortening
depicting a 3-D object such as a double-cube affects
perception of the depicting features on the picture surface.
What could cause this perceptual illusion?
University of Toronto, Scarborough
Results
Results
Projective Theory: the illusion may be caused by projective
Subjects’ responses were converted into Bias scores. If the
The largest biases were again in the double-cube condition
information for the angles of the cube.
response equals the angle-on-the-page, Bias = 0. But if the
(see Figure 4).
Good Form Theory: the angles may be misperceived
response equals the depicted angle, Bias = 1. Intermediate
because they are depicting very standard angles (good
Bias scores reflect the distance of responses from angle-on-
forms), such as 90°.
the-page.
0.8
These two hypotheses can be tested with pictures that
Errors were biased towards each of the depicted angles of
depict acute angles, for example, pictures of a double-cube
the double-cube, not just 90°. The largest biases were in the
presented at various tilts from 5° to 85 ° (see Figure 2).
double-cube condition (see Figure 3). Bias in the V-shape
condition can be explained as due to restriction of Bias
5°
15°
25°
1
Depicted
Angles
0.6
26.6°
63.4°
90°
Bias
Cortical Mechanisms of Vision
June 19 to 23, 2007
0.4
scores to values between 0 and 1.
0.2
1
0
Double-cube
35°
45°
0.8
55°
Depicted
Angles
Introduction
In Figure 1, using parallel projection, the corners of a cube
0.6
26.6°
63.4°
90°
Bias
are depicted by lines that meet at obtuse or acute angles on
the page. What happens when an observer is asked to
0.4
judge these “angles-on-the-page”? Errors are made in the
direction of the depicted angle of 90° (Hammad et al., in
Figure 4. Double-Cube condition Bias scores are higher than Triangle
condition.
Conclusion
The results from both experiments support a Projective
Theory of the illusion: that the foreshortening depicting a 3-
65°
75°
85°
0.2
D object such as a double-cube affects perception of the
press). They may be perceived incorrectly due to the
depicting features on the picture surface (Arnheim, 1977).
0
picture’s perspective (Arnheim, 1977; Gibson, 1979) and
Not only were the errors biased towards “good form” 90°
V-shape
misapplied shape constancy (Gregory, 1972).
Figure 2. Double-cubes at various tilts.
Double-cube
Figure 3. Double-Cube condition Bias scores are higher than V-shape
condition.
depicted angles, but also to 26.6° and 63.4° depicted angles
that are not “good form” angles.
DEDICATION: We dedicate this poster to the memory of
Rudolf Arnheim (July 15, 1904 — June 9, 2007)
X
Figure 1. A double-cube. For angle X, angle-on-the-page = 159° and
depicted angle = 90°
Triangle
Experiment 1
Experiment 2
Subjects judged angles presented in two conditions:
The V-shapes were shown as parts of triangles, to test
1) Double-Cube condition: angles presented as parts of a
whether triangles alone could create the same magnitude of
right-angled, scalene triangle depicted as the top surface of
illusion, perhaps by providing projective 3-D information for
a double-cube (with depicted angle X=90° and depicted
a single triangle being depicted at different tilts.
angles of 26.6° and 63.4°; see Figures 1 and 2).
Subjects judged angles presented in two conditions:
2) V-Shape condition: angles presented alone (i.e., as V-
1) Triangle condition: angles presented as part of a triangle
shapes)
2) Double-Cube condition: same as in Experiment 1.
ACKNOWLEDGEMENTS: S. Bernstein and J. Hockin (TSTOP,
Ontario) for comments, Sandacre Technology for programming, and I.
Abramov, S. Bhasin, S. El Sebae, M. Fazl, B. Haji-Khamneh, E. Hyatt,
G. Ilie, S. Moid, and D. Press.